Reciprocal Cell Calculator

This sheet: https://xraytools.com/reciprocal?view=b

Page: https://xraytools.com/reciprocal

The reciprocal cell of a direct cell: a*, b*, c*, α*, β*, γ* and the reciprocal volume, together with both metric tensors. The reciprocal lattice is where diffraction happens — every reflection hkl is one of its points. The last panel puts the tensors to work on a direction and a plane: how long [u v w] is, how far apart the (h k l) planes are, and the angle between them.

Before this: The reciprocal lattice is a second lattice built from the first, in which each point stands for a family of planes — so a direction in it is a plane normal, not a direction in the crystal. Start from a cell and an index if that sentence is new.
You supply: The six constants of the direct cell. No crystal system and no space group: the reciprocal cell is a function of those six numbers whatever the cell is called. Optionally a direction [u v w] and a plane (h k l), written any way you like — 1 1 0, [110] or 1-10.
Reading it: This page uses the crystallographic convention, a · a* = 1, so a* is 1/d(100) in Å⁻¹; the physics convention puts a 2π in and makes every reciprocal length larger by that factor. Two things a reader expects to be simple are not, and each panel states its own limit where it bites: the starred angles are angles between plane normals, and (h k l) and [u v w] are different kinds of object.

Worked examples: the normal to (100) is 30° away from [100] – quartz · and in a cubic cell they coincide – NaCl, [111] and (111) · the zone law: [110] lies in (110), so hu+kv+lw = 0 · NaCl – halite (rock salt) · Cu – copper, face-centred cubic · α-Fe – ferrite, body-centred cubic · CsCl – caesium chloride · ZnS – sphalerite (zinc blende) · quartz – α-quartz, SiO₂ · cristobalite – α-cristobalite, SiO₂ · berlinite – berlinite, AlPO₄ · rutile – rutile, TiO₂ · aragonite – aragonite, CaCO₃ · ZrO₂ – baddeleyite, monoclinic zirconia · albite – low albite, NaAlSi₃O₈ · urea – urea, CO(NH₂)₂

Earlier on the path: HKL Calculator Next on the path: Space Group Reflection Conditions On From planes to a powder pattern, step 4 of 7

Notation here: d · (hkl) · [uvw] · a*, b*, c* · Z, V — what each one means here

Terms here: zone

What each input changes

(hkl): A plane normal, which is a direction in the reciprocal lattice. In a cell that is not cubic it does not point along the direct-space direction with the same numbers, and seeing that is the point of the page.
[uvw]: A direction in the crystal. Compare it against the same numbers entered as a plane: they coincide only when the metric says so.

Teaching with this page

Objective: After this page a learner can distinguish a plane from a direction with the same three numbers, and say when they coincide.
Start from: this worked example
Ask first: In a hexagonal cell, does the direction [100] point along the normal to the plane (100)?
Watch for: “Yes — they carry the same three numbers”
Then: Space Group Reflection Conditions

Check yourself: In a hexagonal cell, does the direction [100] point along the normal to the plane (100)?

No — only if the metric happens to make them coincide Yes — they carry the same three numbers

The numbers are the same and the objects are not. (hkl) is a plane, and its normal lives in the reciprocal lattice; [uvw] is a direction in the crystal. They coincide in a cubic cell and generally nowhere else — this page draws the angle between them, and in quartz [100] is 30° off the normal to (100). The reason the question names these indices is that the same cell answers the other way for [110] and (110), where the angle is exactly zero: a + b and a* + b* both bisect γ. Coincidence is a property of the indices and the metric together, never of the three numbers alone.

Input

This page uses the crystallographic convention, a · a* = 1, in which a* is measured in Å⁻¹ and is directly the reciprocal of a spacing in Å. Solid-state physics more often writes a · a* = 2π, which makes every reciprocal length here larger by a factor of 2π and every reciprocal volume larger by (2π)³. The starred angles and the shape of the reciprocal lattice are the same in both.

Reciprocal cell

a*	0.23501 Å⁻¹
b*	0.23501 Å⁻¹
c*	0.185007 Å⁻¹
α*	90.000°
β*	90.000°
γ*	60.000°
Cell volume V	113.007 Å³
V*	0.00884898 Å⁻³

Each reciprocal axis is one over a d-spacing: a* is 1/d(100), b* is 1/d(010) and c* is 1/d(001) — the perpendicular spacings of the three families of planes the cell faces lie in. The starred angles are the angles between those plane normals, which is why α* is not in general 180° − α. That holds exactly when the other two angles are right angles — the relation above then collapses to cos α* = −cos α — so it covers the unique angle of a monoclinic cell and γ of a hexagonal one, where 120° gives 60°, and it fails on a triclinic cell and on rhombohedral axes, where no angle is 90°.

Metric tensors

The metric tensor turns Miller indices into a d-spacing without any trigonometry: 1/d² is the row (h k l) times G* times the same column, which is exactly the sum the HKL calculator evaluates. The two are inverses of each other, G* = G⁻¹, and each is symmetric because a dot product does not care which vector comes first.

$G = (\begin{matrix} 24.1415 & -12.0707 & 0 \\ -12.0707 & 24.1415 & 0 \\ 0 & 0 & 29.2162 \end{matrix}) Å^{2}$

$G^{*} = (\begin{matrix} 0.0552299 & 0.027615 & 0 \\ 0.027615 & 0.0552299 & 0 \\ 0 & 0 & 0.0342276 \end{matrix}) Å^{- 2}$

Directions and planes

A plane index and a direction index look alike and are not the same kind of thing. (h k l) names a plane, and lives on the reciprocal basis; [u v w] names a direction, and lives on the direct basis. The normal to (h k l) is the direction [h k l] for every hkl only when the cell is cubic — in any other cell it holds for particular indices, wherever the metric happens to make it so, and this page's own quartz example does exactly that: (1 1 0) against [1 1 0] in hexagonal quartz comes out at 0.000°. The two arrows below are drawn from the same corner at the same length, so the angle between them is the whole difference.

View Default Down a Down b Down c

The [100] direction, solid, and the normal to (100), dashed, both from the origin of one cell. The shaded polygon is where that plane cuts the cell, and the small rings on the axes are where it meets them — those are the intercepts the index is built from, named in the table below. An axis with no ring is one the plane never meets inside this cell. Drag the picture to turn it. Seen from here the two arrows overlap: the angle between them is real, and this viewpoint cannot show it — it looks along the plane the two of them span. Turn it, or take another view.

Length of [100]	4.9134 Å
d(100)	4.25513 Å (1/d = 0.23501 Å⁻¹)
hu + kv + lw	1
Angle to the plane normal	30.000°
Angle to the plane itself	60.000°
Direction along the normal to (100)	[210]
Plane whose normal is [100]	(210)
Miller–Bravais (h k i l)	(1010)
Miller–Bravais [U V T W]	[2110]

A Miller index is defined by where the plane cuts the axes. The plane meets a at a/h, b at b/k and c at c/l, so each index is the reciprocal of an intercept measured in cell fractions — which is the whole reason the indices are reciprocals and not the intercepts themselves. An axis the plane never meets has its intercept at infinity, and 1/∞ is 0: that is what a zero index means, and it does not mean the plane passes through the origin.

Where (100) cuts the axes CSV
axis	index	intercept, in cell fractions	1 ÷ intercept
a	1	1	1
b	0	never — parallel to b	0
c	0	never — parallel to c	0

Turning those intercepts over gives (100), which is the index this panel started from. The scaling only ever goes up. Clear the denominators; do not then divide by a common factor. (1 0 0) and (2 0 0) are different families with different spacings, and the intercept 1/2 has to give 2 — reduced to 1 it would name a family whose planes are twice as far apart as the ones drawn. Reducing by a common factor is how a form is named, which is a different question.

The zone law. A direction lies in a plane exactly when hu + kv + lw = 0, and that sum needs no metric at all: a · a* = 1 and a · b* = 0 by definition, so the dot product of a direction with a plane normal collapses to the products of their indices. Here it is 1, so [100] does not lie in (100); it crosses that plane at 60.000°.

Four indices, because this cell is hexagonal. a = b, γ = 120°, and the three axes at 120° to each other are equivalent while only two of them are in the index — so (100), (010) and (110) are symmetry-equivalent faces whose three-index symbols look unrelated. The redundant third index i = −(h + k) makes that visible. For a direction the conversion is not a subtraction and is the half most people get wrong: U = (2u − v)/3, V = (2v − u)/3, T = −(U + V), W = w, which is why [100] becomes [2110] and not [1000].

How this is calculated

$V = a b c \sqrt{1 - \cos^{2} (α) - \cos^{2} (β) - \cos^{2} (γ) + 2 \cos (α) \cos (β) \cos (γ)}$

$V = 4.9134 Å \times 4.9134 Å \times 5.4052 Å \times \sqrt{0.75} = 113.007 Å^{3}$

Each reciprocal axis is one over the perpendicular spacing of the planes the other two axes lie in — so a* comes from b and c, and the other two follow by cycling a → b → c and α → β → γ together. $a^{*} = \frac{b c \sin (α)}{V}$

$a^{*} = \frac{4.9134 Å \times 5.4052 Å \times \sin (90 °)}{113.007 Å^{3}} = 0.23501 Å^{- 1}$

$b^{*} = \frac{5.4052 Å \times 4.9134 Å \times \sin (90 °)}{113.007 Å^{3}} = 0.23501 Å^{- 1}$

$c^{*} = \frac{4.9134 Å \times 4.9134 Å \times \sin (120 °)}{113.007 Å^{3}} = 0.185007 Å^{- 1}$

The starred angles are the angles between the reciprocal axes, which are the normals to the cell faces. They cycle the same way. $\cos (α^{*}) = \frac{\cos (β) \cos (γ) - \cos (α)}{\sin (β) \sin (γ)}$

$α^{*} = \arccos (\frac{\cos (90 °) \times \cos (120 °) - \cos (90 °)}{\sin (90 °) \times \sin (120 °)}) = 90.000 °$

$β^{*} = \arccos (\frac{\cos (120 °) \times \cos (90 °) - \cos (90 °)}{\sin (120 °) \times \sin (90 °)}) = 90.000 °$

$γ^{*} = \arccos (\frac{\cos (90 °) \times \cos (90 °) - \cos (120 °)}{\sin (90 °) \times \sin (90 °)}) = 60.000 °$

$V^{*} = \frac{1}{V} = \frac{1}{113.007 Å^{3}} = 0.00884898 Å^{- 3}$

Where this comes from

Elucidations on the reciprocal lattice and the Ewald sphere
J. Foadi and G. Evans, Eur. J. Phys. 2008, 29, 1059–1068 · doi:10.1088/0143-0807/29/5/017
Written for somebody learning the reciprocal cell rather than using it: why it is defined the way it is, and what each of its lengths and angles means back in direct space.